We give formulas for integration by parts over the path space and over the loop space of a manifold. We define Sobolev spaces and an Ornstein-Uhlenbeck operator on the loop space. We find some functionals which belongto all the Sobolev spaces.
Invariant Sobolev Calculus on the Free Loop Space
β Scribed by R. Leandre
- Book ID
- 110228673
- Publisher
- Springer Netherlands
- Year
- 1997
- Tongue
- English
- Weight
- 649 KB
- Volume
- 46
- Category
- Article
- ISSN
- 0167-8019
No coin nor oath required. For personal study only.
π SIMILAR VOLUMES
In this paper we will prove the logarithmic Sobolev inequality on free loop groups for various heat kernel measures which P. Malliavin (1989Malliavin ( , 1991, in ``Diffusion Process and Related Problems in Analysis (M. A. Pinsley, Ed.), Vol. I, Birkha user, Basel) constructed. Those measures are as
Taking lim sup L Γ lim k Γ in both sides of (2.10), by (i), (2.8), (2.9), and the fact that lim k Γ \* k =1 we get a contradiction. Hence, , is not identically zero.
We obtain a log-Sobolev inequality with a neat and explicit potential for the gradient on a based loop space over a compact Riemannian manifold. The potential term relies only on the curvature of the manifold and the Hessian of the heat kernel, and is L p -integrable for all p 1. The log-Sobolev ine